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Mirrors > Home > ILE Home > Th. List > imai | GIF version |
Description: Image under the identity relation. Theorem 3.16(viii) of [Monk1] p. 38. (Contributed by NM, 30-Apr-1998.) |
Ref | Expression |
---|---|
imai | ⊢ ( I “ 𝐴) = 𝐴 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | dfima3 4968 | . 2 ⊢ ( I “ 𝐴) = {𝑦 ∣ ∃𝑥(𝑥 ∈ 𝐴 ∧ 〈𝑥, 𝑦〉 ∈ I )} | |
2 | df-br 4001 | . . . . . . . 8 ⊢ (𝑥 I 𝑦 ↔ 〈𝑥, 𝑦〉 ∈ I ) | |
3 | vex 2740 | . . . . . . . . 9 ⊢ 𝑦 ∈ V | |
4 | 3 | ideq 4774 | . . . . . . . 8 ⊢ (𝑥 I 𝑦 ↔ 𝑥 = 𝑦) |
5 | 2, 4 | bitr3i 186 | . . . . . . 7 ⊢ (〈𝑥, 𝑦〉 ∈ I ↔ 𝑥 = 𝑦) |
6 | 5 | anbi2i 457 | . . . . . 6 ⊢ ((𝑥 ∈ 𝐴 ∧ 〈𝑥, 𝑦〉 ∈ I ) ↔ (𝑥 ∈ 𝐴 ∧ 𝑥 = 𝑦)) |
7 | ancom 266 | . . . . . 6 ⊢ ((𝑥 ∈ 𝐴 ∧ 𝑥 = 𝑦) ↔ (𝑥 = 𝑦 ∧ 𝑥 ∈ 𝐴)) | |
8 | 6, 7 | bitri 184 | . . . . 5 ⊢ ((𝑥 ∈ 𝐴 ∧ 〈𝑥, 𝑦〉 ∈ I ) ↔ (𝑥 = 𝑦 ∧ 𝑥 ∈ 𝐴)) |
9 | 8 | exbii 1605 | . . . 4 ⊢ (∃𝑥(𝑥 ∈ 𝐴 ∧ 〈𝑥, 𝑦〉 ∈ I ) ↔ ∃𝑥(𝑥 = 𝑦 ∧ 𝑥 ∈ 𝐴)) |
10 | eleq1 2240 | . . . . 5 ⊢ (𝑥 = 𝑦 → (𝑥 ∈ 𝐴 ↔ 𝑦 ∈ 𝐴)) | |
11 | 3, 10 | ceqsexv 2776 | . . . 4 ⊢ (∃𝑥(𝑥 = 𝑦 ∧ 𝑥 ∈ 𝐴) ↔ 𝑦 ∈ 𝐴) |
12 | 9, 11 | bitri 184 | . . 3 ⊢ (∃𝑥(𝑥 ∈ 𝐴 ∧ 〈𝑥, 𝑦〉 ∈ I ) ↔ 𝑦 ∈ 𝐴) |
13 | 12 | abbii 2293 | . 2 ⊢ {𝑦 ∣ ∃𝑥(𝑥 ∈ 𝐴 ∧ 〈𝑥, 𝑦〉 ∈ I )} = {𝑦 ∣ 𝑦 ∈ 𝐴} |
14 | abid2 2298 | . 2 ⊢ {𝑦 ∣ 𝑦 ∈ 𝐴} = 𝐴 | |
15 | 1, 13, 14 | 3eqtri 2202 | 1 ⊢ ( I “ 𝐴) = 𝐴 |
Colors of variables: wff set class |
Syntax hints: ∧ wa 104 = wceq 1353 ∃wex 1492 ∈ wcel 2148 {cab 2163 〈cop 3594 class class class wbr 4000 I cid 4284 “ cima 4625 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 ax-io 709 ax-5 1447 ax-7 1448 ax-gen 1449 ax-ie1 1493 ax-ie2 1494 ax-8 1504 ax-10 1505 ax-11 1506 ax-i12 1507 ax-bndl 1509 ax-4 1510 ax-17 1526 ax-i9 1530 ax-ial 1534 ax-i5r 1535 ax-14 2151 ax-ext 2159 ax-sep 4118 ax-pow 4171 ax-pr 4205 |
This theorem depends on definitions: df-bi 117 df-3an 980 df-tru 1356 df-nf 1461 df-sb 1763 df-eu 2029 df-mo 2030 df-clab 2164 df-cleq 2170 df-clel 2173 df-nfc 2308 df-ral 2460 df-rex 2461 df-v 2739 df-un 3133 df-in 3135 df-ss 3142 df-pw 3576 df-sn 3597 df-pr 3598 df-op 3600 df-br 4001 df-opab 4062 df-id 4289 df-xp 4628 df-rel 4629 df-cnv 4630 df-dm 4632 df-rn 4633 df-res 4634 df-ima 4635 |
This theorem is referenced by: rnresi 4980 cnvresid 5285 ecidsn 6575 |
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